About

Hyperbolic cosine, denoted cosh(x), is defined as the even part of the exponential function: cosh(x) = (e^x + e^−x) ÷ 2. Unlike trigonometric cosine which oscillates between −1 and 1, cosh is always ≥ 1 and grows exponentially. It describes the shape of a catenary curve: a uniform chain hanging under gravity. Errors in catenary calculations propagate into cable sag, arch stress, and suspension bridge geometry. This calculator computes cosh(x) for arbitrary real x in radians or degrees, provides the inverse acosh, and shows the Taylor series partial sum for verification.

The tool assumes standard IEEE 754 double-precision arithmetic. Results lose significance beyond approximately 15 decimal digits. For |x| > 710 (radians), overflow to Infinity occurs because e⁷¹⁰ exceeds the representable range. Pro tip: when converting degrees to radians, accumulated floating-point rounding means cosh(360°) will not exactly equal cosh(0) due to the intermediate multiplication by π ÷ 180.

Formulas

The primary definition of hyperbolic cosine uses the exponential function:

cosh(x) = e^x + e^−x2

The Taylor series expansion around x = 0:

cosh(x) = ∞∑n=0 x²ⁿ(2n)! = 1 + x²2! + x⁴4! + x⁶6! + …

The inverse hyperbolic cosine for x ≥ 1:

acosh(x) = ln(x + √x² − 1)

Degree to radian conversion applied before computation:

x_rad = x_deg × π180

Key identity linking hyperbolic functions:

cosh²(x) − sinh²(x) = 1

Where x is the input value in radians, e is Euler's number (≈ 2.71828), π ≈ 3.14159, and ln denotes the natural logarithm.

Reference Data

x rad	cosh(x)	sinh(x)	tanh(x)	e^x
0.0	1.000000	0.000000	0.000000	1.000000
0.5	1.127626	0.521095	0.462117	1.648721
1.0	1.543081	1.175201	0.761594	2.718282
1.5	2.352410	2.129279	0.905148	4.481689
2.0	3.762196	3.626860	0.964028	7.389056
2.5	6.132289	6.050204	0.986614	12.182494
3.0	10.067662	10.017875	0.995055	20.085537
3.5	16.572824	16.542627	0.998178	33.115452
4.0	27.308233	27.289917	0.999329	54.598150
4.5	45.014120	45.003011	0.999753	90.017131
5.0	74.209949	74.203211	0.999909	148.413159
6.0	201.715636	201.713157	0.999988	403.428793
7.0	548.317035	548.316123	0.999998	1096.633158
8.0	1490.479161	1490.478826	0.999999	2980.957987
10.0	11013.232920	11013.232875	≈1.000000	22026.465795
−1.0	1.543081	−1.175201	−0.761594	0.367879
−2.0	3.762196	−3.626860	−0.964028	0.135335
−3.0	10.067662	−10.017875	−0.995055	0.049787
−5.0	74.209949	−74.203211	−0.999909	0.006738

Frequently Asked Questions

By definition, cosh(x) = (eˣ + e⁻ˣ) / 2. Both eˣ and e⁻ˣ are strictly positive for all real x. By the AM-GM inequality, their arithmetic mean is always ≥ √(eˣ · e⁻ˣ) = √(1) = 1. The minimum occurs at x = 0, where cosh(0) = 1 exactly.

cos(x) is the trigonometric cosine, periodic with period 2π, bounded between −1 and 1, and defined on the unit circle. cosh(x) is the hyperbolic cosine, non-periodic, unbounded (grows exponentially), and defined on the unit hyperbola x² − y² = 1. They are related by cosh(ix) = cos(x), connecting them via complex numbers.

The maximum representable double-precision float is approximately 1.7976931 × 10³⁰⁸. Since cosh(x) ≈ eˣ/2 for large x, overflow occurs when eˣ/2 exceeds this limit. This happens at |x| ≈ 710.48. For inputs beyond ±710 radians, this calculator will return Infinity and display a warning.

Converting degrees to radians multiplies by π/180. Since π is irrational and IEEE 754 stores it with ~15.9 significant digits, the conversion introduces a small rounding error of order 10⁻¹⁶. For large degree values (e.g., 10000°), this error is amplified by the exponential growth of cosh, potentially shifting the last 2-3 displayed digits.

For |x| ≤ 1, 6 terms (up to x¹⁰/10!) give approximately 15 digits of accuracy. For |x| = 5, roughly 15-20 terms are required. The series converges for all real x, but convergence slows dramatically for large |x|. This calculator shows the first 10 terms of the series expansion for educational comparison against the exact value.

The catenary curve y = a·cosh(x/a) describes hanging cables (power lines, suspension bridges). Errors in cosh directly translate to incorrect sag calculations. In thermal physics, the Langevin function uses cosh for magnetic susceptibility. In special relativity, rapidity addition uses cosh for Lorentz factor computation. Neural network activation functions (e.g., Mish) also employ cosh internally.

This calculator is designed for real-valued inputs only. For complex z = a + bi, the identity cosh(z) = cosh(a)cos(b) + i·sinh(a)sin(b) applies, but implementing complex arithmetic is outside the scope of this tool. The calculator will reject non-numeric inputs and display an error toast.